Abstract
Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via electromagnetic forces.
Keywords
- Sigma Model
- Ward Identity
- Random Environment
- Grassmann Variable
- Quantum Diffusion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
A. Abdesselam, Grassmann-Berezin calculus and theorems of the matrix-tree type. Adv. Appl. Math. 33, 51–70 (2004). http://arxiv.org/abs/math/0306396
E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions. Phys. Rev. Lett. 42, 673 (1979)
E.J. Beamond, A.L. Owczarek, J. Cardy, Quantum and classical localisation and the Manhattan lattice. J. Phys. A Math. Gen. 36, 10251 (2003) [arXiv:cond-mat/0210359]
F.A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987)
H. Brascamp, E. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Func. Anal. 22, 366–389 (1976)
E. Brezin, V. Kazakov, D. Serban, P. Wiegman, A. Zabrodin, in Applications of Random Matrices to Physics. Nato Science Series, vol. 221 (Springer, Berlin, 2006)
F. Constantinescu, G. Felder, K. Gawedzki, A. Kupiainen, Analyticity of density of states in a gauge-invariant model for disordered electronic systems. J. Stat. Phys. 48, 365 (1987)
P. Deift, Orthogonal Polynomials, and Random Matrices: A Riemann-Hilbert Approach (CIMS, New York University, New York, 1999)
P. Deift, T. Kriecherbauer, K.T.-K. McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52, 1335–1425 (1999)
P. Diaconis, in Recent Progress on de Finetti’s Notions of Exchangeability. Bayesian Statistics, vol. 3 (Oxford University Press, New York, 1988), pp. 111–125
M. Disertori, Density of states for GUE through supersymmetric approach. Rev. Math. Phys. 16, 1191–1225 (2004)
M. Disertori, T. Spencer, Anderson localization for a SUSY sigma model. Comm. Math. Phys. 300, 659–671 (2010). http://arxiv.org/abs/0910.3325
M. Disertori, H. Pinson T. Spencer, Density of states for random band matrices. Comm. Math. Phys. 232, 83–124 (2002). http://arxiv.org/abs/math-ph/0111047
M. Disertori, T. Spencer, M.R. Zirnbauer, Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. Comm. Math Phys. 300, 435 (2010). http://arxiv.org/abs/0901.1652
K.B. Efetov, in Anderson Localization and Supersymmetry, ed. by E. Abrahams. 50 Years of Anderson Localization (World Scientific, Singapore, 2010). http://arxiv.org/abs/1002.2632
K.B. Efetov, Minimum metalic conductivity in the theory of localization. JETP Lett. 40, 738 (1984)
K.B. Efetov, Supersymmetry and theory of disordered metals. Adv. Phys. 32, 874 (1983)
K.B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997)
L. Erdős, Universality of Wigner random matrices: A survey of recent results. Russian Math. Surveys. 66(3) 507–626 (2011). http://arxiv.org/abs/1004.0861
L. Erdős, B. Schlein, H-T. Yau, Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287, 641–655 (2009)
L. Erdős, M. Salmhofer, H-T. Yau, Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math. 200, 211 (2008)
L. Erdős, in Les Houches Lectures: Quantum Theory from Small to Large Scales, Lecture notes on quantum Brownian motion, vol. 95, http://arxiv.org/abs/1009.0843
L. Erdős, H.T. Yau, J. Yin, Bulk universality for generalized Wigner matrices. to appear in Prob. Th. and Rel.Fields. [arxiv:1001.3453]
J. Feldman, H. Knörrer, E. Trubowitz, in Fermionic Functional Integrals and the Renormalization Group. CRM Monograph Series (AMS, Providence, 2002)
J. Fröhlich, T. Spencer, The Kosterlitz Thouless transition. Comm. Math. Phys. 81, 527 (1981)
J. Fröhlich, T. Spencer, On the statistical mechanics of classical Coulomb and dipole gases. J. Stat. Phys. 24, 617–701 (1981)
J. Fröhlich, B. Simon, T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking. Comm. Math. Phys. 50, 79 (1976)
Y.V. Fyodorov, in Basic Features of Efetov’s SUSY , ed. by E. Akkermans et al. Mesoscopic Quantum Physics (Les Houches, France, 1994)
Y.V. Fyodorov, Negative moments of characteristic polynomials of random matrices. Nucl. Phys. B 621, 643–674 (2002). http://arXiv.org/abs/math-ph/0106006
I.A. Gruzberg, A.W.W. Ludwig, N. Read, Exact exponents for the spin quantun Hall transition. Phys. Rev. Lett. 82, 4524–4527 (1999)
T. Guhr, in Supersymmetry in Random Matrix Theory. The Oxford Handbook of Random Matrix Theory (Oxford University Press, Oxford, 2010). http://arXiv:1005.0979v1
J.M. Kosterlitz, D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973)
P. Littelmann, H.-J. Sommers, M.R. Zirnbauer, Superbosonization of invariant random matrix ensembles. Comm. Math. Phys. 283, 343–395 (2008)
F. Merkl, S. Rolles, in Linearly Edge-Reinforced Random Walks. Dynamics and Stochastics. Lecture Notes-Monograph Series, vol. 48 (2006), pp. 66–77
F. Merkl, S. Rolles, Edge-reinforced random walk on one-dimensional periodic graphs. Probab. Theor. Relat. Field 145, 323 (2009)
A.D. Mirlin, in Statistics of Energy Levels, ed. by G.Casati, I. Guarneri, U. Smilansky. New Directions in Quantum Chaos. Proceedings of the International School of Physics “Enrico Fermi”, Course CXLIII (IOS Press, Amsterdam, 2000), pp. 223–298. http://arXiv.org/abs/cond-mat/0006421
L. Pastur, M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant matrix ensembles. J. Stat. Phys. 86, 109–147 (1997)
R. Peamantle, Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16, 1229–1241 (1988)
Z. Rudnick, P. Sarnak, Zeros of principal L-functions and random-matrix theory. Duke Math. J. 81, 269–322 (1996)
M. Salmhofer, Renormalization: An Introduction (Springer, Berlin, 1999)
L. Schäfer, F. Wegner, Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes. Z. Phys. B 38, 113–126 (1980)
J. Schenker, Eigenvector localization for random band matrices with power law band width. Comm. Math. Phys. 290, 1065–1097 (2009)
A. Sodin, The spectral edge of some random band matrices. Ann. Math. 172, 2223 (2010)
A. Sodin, An estimate on the average spectral measure for random band matrices, J. Stat. Phys. 144, 46–59 (2011). http://arxiv.org/1101.4413
T. Spencer, in Mathematical Aspects of Anderson Localization, ed. by E. Abrahams. 50 Years of Anderson Localization (World Scientific, Singapore, 2010)
T. Spencer, in Random Band and Sparse Matrices. The Oxford Handbook of Random Matrix Theory, eds. by G. Akermann, J. Baik, and P. Di Francesco. (Oxford University Press, Oxford, 2010)
T. Spencer, M.R. Zirnbauer, Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions. Comm. Math. Phys. 252, 167–187 (2004). http://arXiv.org/abs/math-ph/0410032
B. Valko, B. Virag, Random Schrödinger operators on long boxes, noise explosion and the GOE, http://arXiv.org/abs/math-ph/09120097
J.J.M. Verbaarschot, H.A. Weidenmüller, M.R. Zirnbauer, Phys. Rep. 129, 367–438 (1985)
F. Wegner, The Mobility edge problem: Continuous symmetry and a Conjecture. Z. Phys. B 35, 207–210 (1979)
M.R. Zirnbauer, The Supersymmetry method of random matrix theory, http://arXiv.org/abs/math-ph/0404057(2004)
M.R. Zirnbauer, Fourier analysis on a hyperbolic supermanifold with constant curvature. Comm. Math. Phys. 141, 503–522 (1991)
M.R. Zirnbauer, Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37, 4986–5018 (1996)
Acknowledgements
I wish to thank my collaborators Margherita Disertori and Martin Zirnbauer for making this review on supersymmetry possible. Thanks also to Margherita Disertori and Sasha Sodin for numerous helpful comments and corrections on early versions of this review and to Yves Capdeboscq for improving the lower bound in (4.88). I am most grateful to the organizers of the C.I.M.E. summer school, Alessandro Giuliani, Vieri Mastropietro, and Jacob Yngvason for inviting me to give these lectures and for creating a stimulating mathematical environment. Finally, I thank Abdelmalek Abdesselam for many helpful questions during the course of these lectures.
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Spencer, T. (2012). SUSY Statistical Mechanics and Random Band Matrices. In: Quantum Many Body Systems. Lecture Notes in Mathematics(), vol 2051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29511-9_4
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